This disclosure pertains to optical systems intended for use with a beam of light (e.g., ultraviolet light), charged particles, X-rays, and the like. More specifically, the disclosure pertains to methods and apparatus for obtaining a measurement of beam blur in such optical systems.
Beam blur is a problem that arises in various types of optical systems, especially optical systems that make use of a beam of charged particles such as electrons or ions. In a charged-particle-beam (CPB) optical system, blur is manifest whenever charged particles in the beam ideally intended to be converged at a single point on, e.g., an image plane actually exhibit a significant spread over a certain distance from the point. Such blur can arise from any of various factors. For example, blur can be caused by spherical aberration of the CPB optical system. As a result of spherical aberration, each location of a point on the image plane at which a charged particle (after having passed through the optical system) is incident is a function of the aperture angle distribution of the charged particles as they pass through the object plane.
By way of example, FIG. 4 illustrates an exemplary xe2x80x9cspot diagramxe2x80x9d (or xe2x80x9cscatter diagramxe2x80x9d) of respective destination positions (at an image plane) of 3000 particles propagating through a CPB optical system of a CPB microlithography apparatus. In FIG. 4 the charged particles collectively exhibit blur at the image plane due to spherical aberration. The destination position of each charged particle in the plot is determined by assigning respective computed aberration coefficients to the charged particle. The aberration coefficients of each charged particle are computed according to a probability distribution of the incident angle of the charged particle at the object plane (reticle plane). Chu and Munro, Optik 61:121-145 (1982). In the example of FIG. 4, the incident angle of each charged particle incident on a reticle (at the object plane) is assigned a respective probability in the distribution of incident angles exhibited by the charged particles on the reticle.
Alternatively, for analyzing beam blur, well-known ray-tracing theory can be employed for determining the destination positions of charged particles passing through a CPB optical system. Whenever ray-tracing theory is used in this manner, the manner in which the charged particles propagate under the influence of a particular combination of electrical and magnetic fields is determined by solving an equation of motion for each particle.
A spot diagram or the like as shown in FIG. 4 generally depicts blur as manifest in two dimensions (X and Y dimensions) at the image plane. Alternatively to a spot diagram, in the context of a CPB microlithography apparatus comprising a CPB optical system, blur can be quantified. Quantified blur commonly is defined as the xe2x80x9cfull width at half maximumxe2x80x9d (denoted xe2x80x9cWxe2x80x9d) of a Gaussian distribution of beam intensity along a dimension (e.g., X or Y) in the image plane, typically at an edge of a projected pattern element. At such an edge, blur typically has a Gaussian distribution, wherein W generally is regarded as the portion of the distribution located within the range of approximately 12% to approximately 88% of maximum beam intensity. The range of W can vary (e.g., W can be from 10% to 90%) depending upon the accuracy and precision desired and upon other factors considered in determining blur of the particular system in question. Hence, xe2x80x9cquantified blurxe2x80x9d is distinguished from the comparatively qualitative depiction of blur such as shown in FIG. 4.
Quantified blur is a major factor used in determining the minimum pattern linewidth resolvable by a particular CPB microlithography apparatus. Rapid and accurate quantification of blur is necessary not only when designing an actual CPB microlithography apparatus but also when evaluating a particular reticle pattern to be exposed using the apparatus. For example, quantified blur can be used to produce data concerning any local resizing or the like of the pattern as defined on the reticle, as required for correcting proximity effects.
An exemplary conventional procedure for quantifying blur is based on preparing a grouped tabulation of obtained data concerning destination positions. The grouped data thus are suitable for plotting as a histogram from which blur is quantified. The method comprises the following steps:
(a) A sequence a[i] of xe2x80x9cixe2x80x9d groups (i=1, 2, 3, . . . n) is designated, wherein each group corresponds to a respective range of destination positions xe2x80x9cxxe2x80x9d or xe2x80x9cyxe2x80x9d along a respective dimension (X or Y) in the image plane. In a subsequent step, the respective numbers of individual charged particles incident in the respective range corresponding to each group are tallied. The number n of groups typically is selected based on a tradeoff of calculation accuracy and available time to perform the calculations. The total range is sufficiently large to allow accounting for all the charged particles. By way of example, whenever spherical aberration is the dominant manifestation of beam blur, it is possible to determine, at least roughly, the range of aberration data. For each charged particle, the aberration is calculated by multiplying the aberration coefficient by the third power of the particle""s incident angle. E.g., if the incident angle is limited to 6 mrad and the aberration coefficient is 0.1, then the maximum destination position is (0.1)(6/1000)3=21.6 nm from the convergence point. In this situation, this step (a) is performed before step (b), below.
However, if no prior knowledge exists about the range of data that will be obtained in step (b), then step (b) is performed before step (a). I.e., the data obtained in step (b) is reviewed to determine a suitable number of groups and/or their respective ranges. (Alternatively, the program used to perform steps (a) and (b) can accommodate adding more groups or re-setting group ranges.)
The range of each group is determined based on the required accuracy of the calculations. Hence, if the required accuracy, the total range of the data, and the maximum destination position of the data are known, then n can be readily determined.
(b) A defined number of values of a particular beam parameter (i.e., a parameter that affects blur, such as aperture angle) is selected. The selection is performed either randomly or according to a desired distribution. For example, a randomly selected population of 100 aperture angles is selected. The values are individually substituted into an appropriate function for the particular parameter (e.g., an aberration function) or in a ray-tracing program. The corresponding destination positions of charged particles at the image plane are determined from the substitution calculations or ray traces. If a destination position determined from a single calculation or ray trace falls within a particular destination-position group as designated in step (a), then the tally for that group is incremented by one. The tally data are plotted as a histogram comprising bars corresponding to respective groups.
(c) Step (b) is repeated as required until a satisfactory discernment can be made, from the histogram, of the distribution of the destination positions of the charged particles. The histogram extends in a specific axis direction (e.g., X or Y direction), and the xe2x80x9ccomponentsxe2x80x9d of the distribution (each component corresponding to a respective bar of the histogram) in that axis direction are determined.
(d) For each component of the distribution determined in step (c), a convolution is performed of the distribution with a step function. For example, if the specific axis direction is the X-axis, then the convolution is performed with a step function such that the result of the convolution is 0 at x less than 0 and 1 at xxe2x89xa70. The blur is quantified in the specific axis direction based on the result of the respective convolution. For example, the difference between respective values of x corresponding to 12% and 88% of the maximum value of the convolution function is the quantified blur.
A Gaussian function can be defined in which the quantified blur determined as summarized above is regarded as the full width at half maximum of the distribution. If this function is a point-spread function (PSF), then proximity-effect calculations or exposure calculations can be performed using the PSF.
A specific example of this conventional procedure is as follows. A sequence a[i] (where i=1, 2, 3, . . . 8) of eight groups is designated (n=8). Each group corresponds to a respective range of x coordinates of destination positions, according to the expression (ixe2x88x925)*10xe2x89xa6x less than (ixe2x88x924)*10 (nm). Hence, xe2x88x9240xe2x89xa6xxe2x89xa6xe2x88x9230 for the group in which i=1, and xe2x88x9230xe2x89xa6xxe2x89xa6xe2x88x9220 for the group in which i=2, and so on. Assuming the charged particle beam is an electron beam, each destination position on the image plane is a function of the respective aperture angle of a respective electron in the beam. Twenty-five aperture angles are selected randomly and substituted into an aberration function to yield the respective projections onto the x-axis of the resulting destination positions. The resulting data are listed in Table 1. (The destination positions alternatively can be determined using ray-tracing theory instead of an aberration function.)
In this example, steps (b) and (c), above, yield a determination of the distribution of the destination positions in the x-axis direction for the eight groups i =1,2,3, . . . 8 as shown in Table 2 and plotted in FIG. 1, in which, essentially, the data of Table 1 are sorted into their respective groups and tallied. The convolution (step (d)) yields the data listed in Table 3 and plotted in FIG. 2, in which the data of Table 2 are set forth as cumulative counts. In FIGS. 1 and 2, group numbers (i) are plotted on the abscissa in order of increasing numerical designation, and respective tallies (xe2x80x9ccountsxe2x80x9d) are plotted on the ordinate (the counts in Table 3 and FIG. 2 are cumulative counts). The Gaussian distribution of the data is evident in FIG. 1. FIG. 2 shows the 12% and 88% lines of the cumulative distribution.
The quantified blur is obtained from histogram data situated between the 12% and 88% limit lines shown in FIG. 2, representing the range of 12% to 88% of the maximum value of the convolution function. In FIG. 2, the 12% line corresponds to a cumulative count of 3. Hence, the three most negative destination positions (i.e., xe2x88x9231.7071, xe2x88x9225.1993, and xe2x88x9217.1993) are outside the 12% limit, and the fourth most negative destination position (xe2x88x9211.7665) is inside the stated range. Thus, about half the bar corresponding to i=3 is inside the stated range and about half is outside the stated range. Since the bar corresponding to i=3 covers the range x=xe2x88x9220 to xe2x88x9210, the value of x corresponding to the 12% limit is designated as xe2x88x9215 nm. Similarly, the 88% line corresponds to a cumulative count of 22. Hence, the three most positive destination positions (i.e., 16.94416, 21.13388, and 35.00438) are outside the 88% limit, and the fourth most positive destination position (9.002748) is inside the stated range. Thus, about half the bar corresponding to i=6 is inside the stated range, and about half is outside the stated range. Since the bar i=6 covers the range x=10 to 20, the value of x corresponding to the 88% limit is designated as 15 nm. In summary, regarding this data, the respective corresponding destination positions are:
25* 0.12=3; thus x=xe2x88x9215 nm 
25* 0.88=22; thus x=+15 nm 
The quantified blur is the difference between the two values of x; i.e., 15xe2x88x92(xe2x88x9215)=30 nm.
With the conventional method summarized above, to increase the accuracy and precision of the determined values of blur, the width of each group of the sequence a[i] should be as short as possible (i.e., n should be as large as practicable). Unfortunately, with increases in n, a correspondingly greater number of individual destination positions should be calculated to obtain a meaningful tally in each group. These requirements result in very long computation times, which are impractical or impossible to accommodate especially in modern high-throughput fabrication environments. Also, by calculating blur based on data concerning respective positions and midpoints of histogram bars, a certain inaccuracy of blur calculation inevitably results.
In view of the shortcomings of conventional methods, as summarized above, for quantifying beam blur, the present invention provides, inter alia, methods and apparatus for quantifying blur at high accuracy from a minimal quantity of data.
According to a first aspect of the invention, methods are provided for quantifying blur, of a beam of an xe2x80x9coptical medium,xe2x80x9d exhibited by an optical system through which the beam of the optical medium passes. The xe2x80x9coptical mediumxe2x80x9d in this context can be, for example, light, X-rays, or charged particle beams. In an embodiment of the method, in an image plane of the optical system, respective destination positions of a number (N) of rays of the optical medium are determined. The rays are regarded as originating according to a probability distribution (e.g., a random probability distribution) from respective points in an object plane of the optical system. The destination positions are projected onto an axis in a direction, in the image plane, in which blur is to be quantified. Respective coordinates of each of the destination positions on the axis are determined. The determined coordinates are ranked. From the ranking numbers and the respective determined coordinates, blur is calculated.
The destination positions can be determined from respective ray traces of the optical medium extending from points on the probability distribution and through the optical system to the respective destination positions on the image plane. Alternatively, the destination positions can be determined according to respective solutions of an aberration function.
Blur desirably is quantified as a region between a predetermined lower limit and a predetermined upper limit. By way of example, the lower limit and upper limit can be determined so as to include, in the calculation of blur, only destination-position data located between a lower limit of 12% and an upper limit of 88%.
Continuing further with the example, the lower limit can be determined to exclude, from the calculation of blur, 12% of the total number of destination positions that are most negative. Similarly, the upper limit can be determined to exclude 12% of the total number of destination positions that are most positive.
Blur can be calculated as respective coordinates of respective destination positions corresponding, in the ranked data, to a predetermined lower limit and a predetermined upper limit.
According to another aspect of the invention, apparatus are provided for quantifying blur, of a beam of an optical medium, exhibited by an optical system through which the beam of the optical medium passes. An embodiment of the apparatus comprises means for determining, in an image plane of the optical system, respective destination positions of a number (N) of rays of the optical medium originating according to a probability distribution from respective points in an object plane of the optical system. The apparatus also includes means for calculating respective coordinates of the destination positions on a projection axis, in the image plane, in which blur is to be quantified. The apparatus also includes means for ranking the coordinates, and means for calculating blur from the ranked coordinates.
With methods and apparatus as disclosed herein, the actual data regarding ranked destination positions are used rather than data pertaining to a histogram of such data. Even though histograms are useful for certain data-analysis purposes, the act of converting data to a histogram inevitably tends to bury the data. By using the destination-position data directly, the inherent inaccuracies imposed by creating a histogram are not introduced. Thus, blur is calculated more quickly and more accurately using less data.